Hod–Piran conjecture on decay of charged scalar fields

Let $q_0$ be the scalar-field charge, $e$ the asymptotic black-hole charge at timelike infinity, $m$ the scalar-field mass,

$$\mathcal{Q}^+$$

the relevant quotient spacetime,

$$\mathcal{I}^+$$

future null infinity,

$$\mathcal{H}^+$$

the event horizon, and $v$ a null coordinate fixed by the stated gauge choice. Define

$$\delta(q_0e):=1-\operatorname{Re}\left(\sqrt{1-4(q_0e)^2}\right)\in[0,1).$$

Hod–Piran conjecture. Among all admissible, sufficiently regular and decaying data, there exists a generic subclass such that, whenever the maximal future development satisfies $\mathcal{Q}^+\cap J^-(\mathcal{I}^+)\neq\emptyset$, in the charged massless case $q_0\neq0$ and $m^2=0$,

$$|\phi|_{|\mathcal{H}^+}(v)\sim v^{-2+\delta(q_0e)},\qquad |D_v\phi|_{|\mathcal{H}^+}(v)\sim v^{-2+\delta(q_0e)}.$$

This is a heuristic late-time tail prediction supported by numerical evidence; only an upper bound has been recovered rigorously in a restricted fixed-background, small-charge regime, so the generic nonlinear assertion remains open.